Problem: Determine how many solutions exist for the system of equations. ${-6x+y = 9}$ ${12x-2y = -10}$
Solution: Convert both equations to slope-intercept form: ${-6x+y = 9}$ $-6x{+6x} + y = 9{+6x}$ $y = 9+6x$ ${y = 6x+9}$ ${12x-2y = -10}$ $12x{-12x} - 2y = -10{-12x}$ $-2y = -10-12x$ $y = 5+6x$ ${y = 6x+5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x+9}$ ${y = 6x+5}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.